Retractable Air Beam Roof for Large Stadiums

ABSTRACT

The detailed principle, structure, manufacture and designing method of a retractable roof for large stadiums or other huge spaces are established. This roof is formed by using a number of connected and paralleled inflatable arc shaped tubes (called “air beams”) made of soft material (membrane). Using successive inflation to the tubes can realize deploying of the roof, and using successive deflation of tubes can realize retraction of the roof. All these processes can be completed by an automatically controlled program. A rigorous mathematics derivation and mechanics proof show that, the retractable air beam roof with durable membrane of ordinary durability can have a large span of over 100 meters, and can withstand in a regularly extreme (critical) weather (heavy wind, rain, snow and hail). The outstanding strong point of this kind of retractable roof is absolutely safe for spectators and performers (or competitors) inside the stadium.

SUMMARY

1. The concept of that, an inflatable arc shaped tube of soft material (membrane) can be used as a beam structure to hold a certain amount of load, is initiated (Jun. 2, 2006, Filed at National Patent Office of China, File No. 200610083730.8, titled “Air Beam and Inflatable, Retractable Mobile Roof for Large Spaces.”)

2. Rigorous mathematics derivation, mechanics proof, and quantitative analysis are completed for a single air beam along with its supporting structure, and for the connected air beams. The calculation formulas or the quantitative relationships between the beam span, beam diameter, arc degree, wind speed and wind direction, thickness of accumulated snow and hail, the air pressure inside the tube, the tension stress in the membrane, and the durability intensity of membrane material are obtained.

3. The outline structure of the retractable roof by using a number of paralleled and connected air beams is established. The method of deploying roof by successively inflation of air beams, and the method of retraction of roof by successively deflation of air beams and by winding chains are invented.

4. Detailed structure, manufacture and design method for all portions of the entire retractable air beam roof, including air beam membrane system, beam ends and sliding track system, inflation and deployment system, deflation and retraction system, electrical, electronic, automatically control system, especially the wind resistance system, are invented.

In comparison with the existing retractable roof design methods and patents, this invention has, at least, the following obvious strong points:

1. The main material of the entire roof is membrane, instead of metal or other solid and heavy materials. Especially, the entire top portion of the roof is made of only soft materials, and without any metal. So, it is light in weight. Moreover, the material is non-flammable. So, it is absolutely safe for both spectators and performers (competitors) inside the stadium.

2. The manufacturing process of the roof is easily attainable and not time consuming.

3. The cost of the entire roof construction, for both the material and labor, are relatively low.

4. Operation and maintenance of the roof is easy and also low cost.

5. The operation of the roof does not affect the air pressure and air circulation for the entire space in the stadium. In other words, the entire retractable roof does not affect the environment inside the stadium.

6. The deploying and retracting of the roof can be easily proceeded without noise or noticeable effects to the spectators and audience.

7. The entire roof construction process does not affect the original stadium structure and the existing stadium operation, and does not affect the environment and neighborhood.

FIELD

By the title of the invention, the field seems to be architectural or civil engineering. In fact, by the content and the academic substance of this invention, its field should be mechanical engineering or membrane material engineering, based on a multi-disciplinary academic area, including mathematics, physics, dynamics, statics, air dynamics, fluid mechanics, electrical engineering, electronic engineering, and material engineering.

STATEMENT OF PRIOR DISCLOSURES BY INVENTOR

From the year 2001, the year of that, City Beijing was approved to sponsor the 2008 Olympic Game, inventor Keh C. Liu started the research and invention work for the Retractable Air Beam Roof for Large Stadium. On Jun. 2, 2006, Liu formally filed his completed invention, titled

“Air Beam and Inflatable, Retractable Mobile Roof for Large Spaces”

at the

National Intellectual Property Patent Office of P. R. China, Beijing,

with File number:

-   -   File No: 200610083730.8

Name of Inventor: Keh Chung Liu Name of Applicant: Li, Qing-li

Please see the submitted copies of the official sealed document:

1. Notice of Acceptance of Patent Application.

2. English Information of Invention Patent Application.

3. Notice of Application Fee.

4. Detailed Description of Invention. (20 pages)

5. Detained Drawings (26 Figures)

The present application is simply an English language translation of the above Chinese Application, with a limited revision, and without any change of the subject matter facts. A little bit change is the mechanical structure at the two ends of each air beam, which is another adoption, and does not affect the entire invention.

DESCRIPTION OF RELATED ARTS

1. An invention entitled “Arch Supported Retractable Inflatable Roof” by Kenneth C. Logan, with U.S. Pat. No. 4,738,057, Patent Date: Apr. 19, 1988, File Date: Apr. 2, 1987 is the closest invention by titles, as far as the inventor knows. The outline of this invention is as follow:

The entire suggested stadium roof consists of 10 or 12 pieces; each piece can slide along a track to realize the retraction (partially retraction). Each piece is made of membrane (fabric) and steel, and inflatable. For retraction, the eight pieces slide into a space under the bigger arch shaped piece to realize the partial retraction. There exists some steel structure at the top of the roof.

The principle and structure of our retractable air beam roof is completely different. The entire top portion of our roof, is made of only soft membrane (fabric) material, without any metal parts. In our invention, the roof can be completely retracted until completely invisible for the spectators and the performers. In our invention, the entire structure of the roof and its inflated parts, are completely different. The inflation and deflation processes, and the retraction and deployment processes are also completely different.

2. Invention titled “Inflatable Vault which be Opened out and Collapsed” by Dalamare, U.S. Pat. No. 5,303,5116, patent Date Apr. 19, 1994, Filed date Jun. 4, 1992: The suggested roof is also inflatable and retractable, but the structure and principle of retraction and deployment in our invention are completely different with this invention, and our roof can cover much larger area.

3. Comparison between Inventions by Brown, Stafford, and others with Liu′ invention is in the following table:

Comparison of Liu's Work with Brown's, Stafford's and Others' Work

Inventor Liu Brown Stafford Gundling Filed Date Jun. 2, 2006 Feb. 12, 2008 Mar. 13, 2018 Jun. 28, 2019 China Patent O USPTO USPTO USPTO Word “Air Beam” yes yes yes yes used Method of using arc yes yes no no shaped inflatable air beam as part for load Method of using Math, yes no no no Dynamics, statics derivation to Obtain formulas for Air Beams Method of using a number of yes no no no Paralleled connected air Beams to form a roof of large spaces Detailed Method and structure yes no no no of retraction and deployment of An air beam roof Detained method, principle yes no no no and design for smoothly retraction and deployment Method of rigorous math and yes no no no Phys proof for abilities of an Air beam roof in bad weather

BRIEF DESCRIPTION OF DRAWINGS

FIG. 1. Dynamics and Statics Analysis for a Single Air Beam with Two Ends

FIG. 2. Dynamics and Statics Analysis for a Single Air Beam with Two Ends under Strong North Wind

FIG. 3. Dynamics and Statics Analysis for a Single Air Beam with Two Ends at the Cross Section Surface as the Imaginary Boundary

FIG. 4. Dynamics and Statics Analysis for a Single Air Beam without Two Ends and with the Cross Section Surface as the Imaginary Boundary during Strong North Wind

FIG. 5. Hard Nylon Reinforced Plate at End of Air Beam (View from the End Portion Moving Direction)

FIG. 6. Nylon Reinforced Plates at End of Air Beam with Nylon Reinforced Membrane and Steel Wheel

FIG. 7. Arranged Air Beams Form Entire Roof Along the Steel Track in Deployed Status (Side View from Horizontal Direction)

FIG. 8. Entire Air Beam Roof in Retracting and in Deploying Processes (Side View from Horizontal Direction)

FIG. 9. Completely Retracted and Deflated Air Beam Roof Stacked at Both Ends of the Stadium (or on the Existing Ring Roof above the Stands)

FIG. 10. Retracted, Deflated, and Stacked Air Beam Roof (Downward View from Top)

FIG. 11. Air Beam Roof in Deployed Status (Downward View from Top)

FIG. 12. Air Beam Roof in Deploying Process and in Retracting Process (Downward View from Top)

FIG. 13. Inflated, Deployed Air Beam Roof in Working Status (Side View from the End in Horizontal Direction)

FIG. 14. Deployed and Inflated Air Beam Roof with Rain Strips Installed (Side View from Horizontal Direction)

FIG. 15. Cross-Section of Air Beam #1 and #2 with Rain Gutters, Rain Strips, and Nylon Reinforced Hard Fiberglass Strips (Side View from Horizontal Direction)

FIG. 16. Steel Block with Pulley and Sliding Ditch Installed at the End of Each Air Beam (View from Wheel Moving Direction)

FIG. 17. Cross-Section of Track along which Air Beams Slide (View from Horizontal Direction of Wheels Moving)

FIG. 18. Steel Block at the End of Air Beam Sliding along Track (Side View from 30° Upward)

FIG. 19. Angle Steel Frame for Supporting Sliding C Channel and Sliding Bar for the End of Air Beam (View from the Moving Direction of End of Air Beam)

FIG. 20. Angle Steel Frame for Supporting the Sliding Bar and Sliding C Channel for the End of Air Beam (Side View from Horizontal Direction Outside of Stadium)

FIG. 21. Bent Ending Portion of the Sliding Track and Sliding Bar along with its Supporting Frame and Sliding Steel Blocks in Retracted Position (Side View from Horizontal Direction)

FIG. 22. Three Dimensional View of Entire Retractable Roof Formed by Parallely Arranged Air Beams in Deployed Working Status

FIG. 23. Retractable Air Beam Roof Built above the Opening of the Existing Ring Roof Structure above Stands of the Stadium (Three Dimensional View in Deployed Status)

FIG. 24. Air Beam Roof in Retracting Process and in Deploying Process, along with the Stadium, above which the Air Beam Roof is Built (Three Dimensional View)

FIG. 25. Air Beam Roof in Near Completion of Retracting Process and in Deploying Process, along with the Stadium, above which the Air Beam Roof is Built (Three Dimensional View)

FIG. 26. Completely Retracted Air Beam Roof Stacked Neatly on the Existing Ring Roof Structure of the Stadium (Three Dimensional View)

FIG. 27. Detailed Deploying Process and Retracting Process of the Air Beam Roof (Three Dimensional View)

FIG. 28. Detailed View of Retracted and Deflated, Neatly Stacked Air Beams on the Ring Roof of Stadium or on the Ground

FIG. 29. Entire End Portion of the Air Beam, Including Hard Nylon Reinforced Plate, Steel Block, Steel Wheel, Sliding Ditch, and Rain Resistant Rope Connection Ring

FIG. 30. External Forces Exerted on the End Portion of the Air Beam (View from the Wheel Moving Direction)

FIG. 31. Statics Analysis for the Membrane Stress and Air Pressure for One Section of the Air Beam (Three Dimensional View)

DETAILED DESCRIPTION OF DRAWINGS

FIG. 1: This figure is for the mathematical derivation and the physics (statics) analysis for a single arc shaped air beam (2). We suppose that, it is in the case of accumulated snow covered, and wind from the North. We assume that, the stadium (50) (the long side of the stadium) is located in the East-West direction. So, the arc shaped air beam crosses from North to South. We take a single air beam as the object in consideration. We also take the force components in only the two dimensions as North-South and Up-Down directions in consideration in the first stage. In this case, all the external forces exerted onto the object will include the follows:

1. W is the resultant weight (gravitational) force due to the selves weight of the tube membrane along with the two end structures.

2. W_(r) is the resultant weight force due to the rain strip membrane (the two portions above this air beam),

3. W_(s) is the resultant weight force of snow accumulated on this air beam and two portions of rain strip,

4. F_(w) is the resultant wind force in the direction of incline down angle as shown.

5. N_(n) and N_(s) are the vertical and horizontal components, respectively, of the forces from the sliding track exerted onto the end of the tube at the North side.

6. F_(n) and F_(s) are the vertical and horizontal components, respectively, of the forces from the sliding track exerted onto the end of the air beam at the South side.

7. L is the length of the air beam.

8. L is the span of the air beam.

9. D is the diameter of the air beam.

FIG. 2: This figure is also for the Math derivation and phys. analysis. All the symbols are the same as FIG. 1. Some parts explicitly drawn are:

1. Eight wind resistant ropes are in the figure (52).

2. τ_(s) is the resultant force of the four tension forces by the four ropes at the South side. This force is from South to North, inclined down for some degree, directed to the North end of the air beam. However, when we consider the case that, the two ends of the air beam are in the considered object, the resultant rope tension forces will be the internal force.

3. The wind lines (4), which show the wind direction and intensity (or speed) distribution, are explicitly drawn.

FIG. 3. This figure is also for Math derivation and Phys analysis. The difference with FIG. 1, is that, in statics analysis, the object considered does not consist of the two end portions of the air beam. Therefore the external forces will contain the following forces instead of the previous N_(n), N_(s), F_(n), and F_(s):

1. N_(n) and N_(s) are the vertical components of the resultant pressure forces at the tube cross section from the air inside the end portion exerted onto the portion of air inside the air tube (5) located at the North end and the South end of the air beam (2), respectively.

2. F_(n) and F_(s) are the horizontal components of the resultant pressure forces at the tube cross section from the air inside the end portion exerted onto the portion of air inside the air tube (5) located at the North end and the South end of the air beam (2), respectively.

3. M_(n) and M_(s) are the resultant forces exerted to the membrane of tube (5) from the two ends structures at the North side and the South side respectively.

FIG. 4. This figure is also for the Math derivation and Phys analysis. It is similar with FIG. 3. In this analysis, the wind forces are considered. In this figure, the considered object does not contain the two ends of the air beam. Therefore, the resultant rope tension force _(s) will be included in the external forces, and be taken into calculation. All external forces N_(n), N_(s), F_(n), F_(s), M_(n), M_(s), F_(w), τ_(n), τ_(s), W, W_(r), and W_(s) are similarly defined as the FIG. 3 and FIG. 2.

FIG. 5: At the two ends of each air beam, there are sophisticatedly designed cone-shaped structures. Each structure, called “End” of Air Beam (18), contains two pieces of (hard) solid nylon reinforced plates (7 and 12). This figure shows one plate (7). The upper edge of the plate is firmly connected with the lower edge of the membrane of the air tube (5). The lower end of the plate (7) will join with a Steel Block (13) by a joining Hinge (15). There is a Vent Valve (8) installed on one plate (7). Among the four plates for each air beam, there is only one vent valve installed in one plate for the purpose of inflation and deflation for this air beam. Each vent valve is controlled by a Motor (9) for opening and closing operation. Each motor has a separate wire connected to the central control table. During the inflation and deflation of the air beam, each pair of plates will move around the hinge, acting similarly to opening and closing a book.

FIG. 6: This figure shows the end structure (18) viewed in another direction, viewed from south to north, with a 30 degree downward incline, and from outside of the sliding track (16). The parts shown include:

i) a nylon reinforced plate (7), which has a vent valve (8) and a motor (9) connected with it,

ii) a nylon reinforced plate (12) without vent valve and without motor connected, (the two plates can be opened to a 50 degree angle when fully inflated)

iii) durable Nylon Reinforced Membrane (10), which is firmly connected with the membrane of the air tube (5) on the upper edge and firmly connected with the two nylon reinforced plates (7 and 12) at the lower edge, and sealed to prevent air leakage, and foldable towards the inside along with the air inflation and deflation.

iv) Steel Block (13) and a Steel Wheel (35) connected to the lower end of the two plates (7 and 12) with two hinges (15).

FIG. 7: This figure shows the side view of the air beam roof formed by parallely connected 48 (supposedly) air beams (19), viewed from South to North, and from the outside. It is in inflated and deployed status. The air beams are divided into two groups. The stadium is supposedly located in the East-West direction. The Western group of air beams consists of beams No. 1, 3, 5, . . . , 47; and the Eastern group consists of beams No. 2, 4, 6, . . . , 48. On the front two air beams of the two groups, beams No. 1 and No. 2, a Rain Gutter (17) is installed on each. In fully inflated status, the two gutters are coupled together for rain proof (see FIG. 15 for details).

FIG. 8: This figure shows the side view (from Southern side to North) of the retraction and deployment processes of the air beams. During the retraction process, the successive deflation process is according to the order: starting from the last air beam in each group, that is beam No. 47 and 48 in present figure, and then, No. 45, 43, 3, 1, for the Northern group, and in same time, No. 46, 44, 4, 2, for the Southern group. In the retraction process, each group of air beams gradually moves apart from each other.

In the deployment process, the order of successive inflation of the air beams is: starting from the front air beams No. 1, then No. 3, 5, . . . , 45, 47, for the Northern group, in the same time, starting from No. 2, then No. 4, 6, . . . , 46, 48, for the Southern group. During the deployment process, the two groups of air beams gradually move toward each other.

FIG. 9: This figure shows the completed status of the retraction process. Both two groups of air beams are deflated and stacked orderly at the two sides of the stadium (or on the existing roof structure 50), which are non-visible for the spectators and performers (players).

FIG. 10: This figure shows the top view of the entire structure of the air beam roof in retracted status.

FIG. 11: This figure shows the top view of the entire air beam roof in inflated and deployment status.

FIG. 12: This figure shows the top view of the entire air beam roof during the process of retracting and the process of deploying.

FIG. 13: This figure shows the side view from the East or from the West (from outside) of the air beam roof in the deployment status or during the processes of deploying and retracting.

FIG. 14: This figure shows the side view from outside, South to North, for the entire air beam roof. The difference between this figure and FIG. 7, is that, this figure depicts the rain strips (27), which are firmly adhering to the air beams' surface and are located in between the air beams for the purpose of protection of rain and snow.

FIG. 15: This figure shows the amplified cross section view of the two air beams No. 1 (30) and No. 2 (31), the front air beams of the two groups, in the status of deployment. The figure also shows the upper rain gutter (28) and lower rain gutter (29), which are in inflated and coupled status. The install location of the rain strips (27) and the solid nylon reinforced fiberglass strips (41) for wind resistant purposes, which are solid and arc shaped with the same span as the air beam, are also shown.

FIG. 16: This figure shows the steel parts connected with the end portion of each air beam, viewed from East to West, at the Southern end of the air beam. These parts include a steel block (13), a steel wheel (35), a ball bearing (36), and a set of cylindrical roller bearing (34), which forms a sliding ditch for sliding along the sliding bar (46 in other figures). The upper end of the steel block (13) is connected to the nylon reinforced plates (7 and 12) through two hinges (15).

FIG. 17: This figure shows the cross section of the sliding track, which consists a C channel (38), a lower rail of 22 kg/m (40), an upper rail of 12 kg/m (37), between which the steel wheel (35) can freely roll, along with the supporting structure made of angle steel frame (39). This figure is also viewed from East to West, at the Southern end of air beam.

FIG. 18: This figure shows the detailed mechanical structure of the sliding mechanism and parts. The figure is viewed from South to North, 30 degree downward incline. The parts include: a steel block (13), a steel wheel (35), cylindrical roller bearing (34), which contains a number of cylindrical rollers (9 rollers in the figure), two nylon reinforced hard plates (7 and 12), fabric nylon reinforced membrane (10), Sliding Bar (46), sliding track (16), which consists of a C channel (38), an upper rail of 12 kg/m (37), a lower rail of 22 kg/m (40),

FIG. 19: This figure shows the cross section of the sliding structure at the end of each air beam, along with its supporting structure made of angle steel of size 10 (39). The sliding structure includes the steel wheel (35), steel block (13), sliding bar (46), sliding track (16), which consists of the C channel (38), the upper rail (37) and the lower rail (40). The supporting structures are usually installed on the existing ring shaped roof structure above the stands of the stadium (50). This figure is viewed from East to West, and in horizontal direction.

FIG. 20: This figure shows the sliding structure and its supporting structure, viewed from South to North, and in horizontal direction from outside. The sliding structure consists: i) a C channel of size 38 cm (38), which is for the installation of the two rails (37 and 40) and for the sliding of the steel wheel (35), ii) a C channel of 14 cm (44), which is for installation of the sliding bar (46). The sliding bar is for the sliding and for keeping correct orientation of the steel block (13). The entire supporting structure frame (53) is made of angle steel of size 10 (39) and angle steel of size 9 (42), and is installed on the top of the existing ring shaped roof above the stands of the stadium (50).

FIG. 21: This figure shows the end portion of the sliding structure, consisting of the sliding track (16) and the sliding bar (46). Both the sliding track and the sliding bar are circular arc shaped at the end portion, and supported by a structure of angle steel (39). In this figure, the air beam roof is in deflation and retraction status. All the end portions (18) of the 24 air beams in the Northern group are stacked orderly along this circular arc shaped sliding track. The figure is viewed from South to North, in horizontal direction and from the outside.

FIG. 22: This figure shows the three dimensional view of the entire air beam roof (49) in deployment and inflated status. It shows the air beams (19) and the sliding track (16).

FIG. 23: This figure shows the three dimensional view of the entire air beam roof (49) constructed above the opening area of the existing ring shaped roof above the stands of a large stadium (50). Part No. 16 is the sliding track. The figure shows the inflated and deployment status of the roof.

FIG. 24: This figure is the three dimensional view of the entire air beam roof (49) constructed above the opening area of the existing ring shaped roof above the stands of a large stadium (50). Part No. 16 is the sliding track. The figure shows the process of inflating and deploying and the process of deflating and retracting.

FIG. 25: This figure is the three dimensional view of the entire air beam roof (49) constructed above the opening area of the existing ring shaped roof above the stands of a large stadium (50). Part No. 16 is the sliding track. The figure shows the beginning stage of the inflating and deploying process or in the completing stage of the deflating and retracting process of the air beam roof.

FIG. 26: This figure is the three dimensional view of the entire air beam roof (49) constructed above the opening area of the existing ring shaped roof above the stands of a large stadium (50). Part No. 16 is the sliding track. The figure shows the status of the roof in completed retraction. The two groups of air beams are completely deflated and orderly stacked on the top of the ring roof.

FIG. 27: This figure shows the amplified FIG. 24. It is also in three dimensional view. The status of the roof is in the deflating and retracting process, or in the inflating and deploying process. It clearly shows the upper rain gutter (21) installed on the air beam No. 1 (30) or the lower rain gutter (22) installed the air beam No. 2 (31), the wind resistant nylon ropes (52), the sliding track (16), the supporting structure frame (53), and the opening of the existing ring roof above the stands of the stadium (24).

FIG. 28: This figure shows the amplified FIG. 26. It is also in three dimensional view. The status of the roof is in completed deflation and retraction. All the air beams in one group are orderly stacked (20). The figure also clearly shows the rain gutter (17) installed on the air beam No. 1 (30) or air beam No. 2 (31), the wind resistant ropes (52), the sliding track (16), the supporting structure frame (53), the opening of existing ring roof above the stands of the stadium (24), the solid nylon reinforced fiberglass strip (41), and the deflated end portion of the air beam (18).

FIG. 29: This figure shows the connection of the end portion of an air beam (in FIG. 5) and the steel block (in FIG. 16). It is also another direction view of FIG. 6, i.e. the view from East to West, from horizontal direction. All the parts have been clearly shown in the two FIGS. 5 and 16. Here they will not be repeatedly described.

FIG. 30: This figure is for Math derivation and statics analysis use for the end structure portion of the air beam. In this analysis, the end portion is taken as the considered object. All the external forces exerted onto this portion are shown in this figure. They are as follows:

1. The distributed air pressure force from the upper portion of air to the lower portion of air at the cross section, which is centralized as the vertical component N′_(n) and the horizontal component F′_(n),

2. The distributed force exerted from the upper membrane to the edge of this end structure (part solid edge, part membrane edge), which is centralized as M_(n).

3. The gravitational force due to the weight of the entire end portion as shown in the figure as W_(e).

4. The vertical and horizontal components of the force exerted from the sliding track to this end portion N_(n) and F_(n).

5. The resultant tension force from the four wind resistant ropes _(s).

All these forces are shown in the figure.

FIG. 31: This figure shows the relationship between the air pressure inside the air tube and the tension stress of the tube membrane for the imaginary cut section of the air tube, including the cut part of air, from which the equilibrium equations can be obtained. It is a convenient method for the purpose of Math derivation and statics analysis for obtaining the relations, equations, and formulas about the abilities of an air beam.

LIST OF PARTS WITH NUMBERS IN DRAWINGS

-   1—Nylon Reinforced Membrane -   2—Arc Shaped Inflatable Membrane Tube (Air Beam) -   3—End Portion of Air Beam (Cone Shaped Inflatable) -   4—Wind Lines -   5—Air Tube -   7—Nylon Reinforced Solid Plate (with Vent Valve and Motor) -   8—Vent Valve -   9—Motor for Driving Vent Valve -   10—Fabric Nylon Reinforced Membrane Material -   12—Nylon Reinforced Solid Plate (with Vent Valve) -   13—Steel Block -   15—Hinge -   16—Sliding Track -   17—Rain Gutters -   18—End Portion of Air Beam -   19—Air Beam -   20—Deflated Stacked Air Beams -   21—Upper Rain Gutters -   22—Lower Rain Gutter -   23—Inflated Air Beam -   24—Opening Edge of Existing Ring Shaped Roof above Stands of Stadium -   25—Top of Existing Ring Shaped Roof above Stands of Stadium -   26—Supporting Structure Frame -   27—Rain Membrane Strip -   28—Upper Rain Gutter Inflated -   29—Lower Rain Gutter Inflated -   30—Air Beam #1 -   31—Air Beam #2 -   32—Half Circular Ring for Wind Resistant Rope Connection -   33—Sliding Ditch -   34—Cylindrical Roller Bearing -   35—Steel Wheel -   36—Ball Bearing -   37—Rail of Size 12 kg/m -   38—C Channel of Size 38 -   39—Angle Steel of Size 10 -   40—Rail of Size 22 kg/m -   41—Solid Nylon Reinforced Fiberglass Arc Shaped Strip -   44—C Channel of Size 14 cm -   46—Sliding Bar -   47—Sliding Structure at Ends of Air Beams -   49—Retractable Air Beam Roof -   50—Existing Ring Shaped Roof Structure above Stands of Stadium -   52—Wind Resistant Nylon Rope -   53—Supporting Structure Frame

INTRODUCTION (BACKGROUND)

In the events of using large stadiums and other huge structures, people wish to have an open-air environment in nice weather. On the other hand, during bad weather, say rain, storm and snow, people wish to have a protective roof to keep the events and performances from being discontinued by the weather. Therefore, to have a retractable roof is critical and desirable at all times.

Automatically retractable roofs for large spaces, such as stadiums and fields, have great importance and necessity. In the present, existing designs and patents for retractable roofs are basically in the following two categories:

1) Huge, heavy sliding blocks. This kind of retractable roof uses steel structure and other heavy materials to make huge blocks, sliding above the stadium to realize the retractable roof.

2) Very high towers located outside of the stadium or in the center of the stadium, from which long cables hang the large roof structure to realize the retractable roof.

For the first kind of roof, due to the large span of the stadium, the size of the blocks reach hundred meters, so the entire block will reach at least many thousand tons. Such a heavy, huge block sliding above the people's heads is tremendously hazardous. Any small factor will cause accidents to happen. So, this kind of roof will cause people to have constant emotional distress. Secondly, this kind of roof is extremely costly to build, operate and maintain.

For the second kind of roof, due to the high tower and long cables, the similar hazardous and distressful situation exists. It also needs high costs to build, operate, and maintain.

Therefore, to have a new type of roof, which is light, safe, and low priced, is absolutely necessary. We found that, air-inflated, arc-shaped tubes can bear certain amounts of load, and can have over a hundred meters span. After rigorous mathematical and mechanical analysis, we found that, air-inflated tubes of certain size and regular durability of membrane will be able to constitute the basic part of the roof with a huge span and withstand the regular bad weather. This kind of roof, we call an air beam roof.

The air beam roof will overcome the shortcomings of the above types of existing retractable roof designs and patents. It will also be able to easily realize the automatically deploying and retracting processes. Because the main material is lightweight membrane, the entire roof is lightweight and absolutely safe. So, the above hazardous situation no longer exists in our invention. It is also low cost in construction, operation, and maintenance. The players and spectators below the roof will feel comfortable and without distress, and will not be affected at all by the operations of the roof.

OUTLINE OF DESIGN (ROUGH DESCRIPTION)

1. The basic part of this roof are arc shaped tubes, which are paralleled and connected, side by side, and can be inflated with air. When all tubes are fully inflated successively, these paralleled and side by side tubes just cover the entire field, and therefore, they become the roof. Therefore, we call these tubes “air beams”.

2. Based on our analysis, the degree of the arc of the air beams is chosen to be 120° to reach the best results. The length of the tube can be 100˜200 meters, which depends on the size and requirement of the field. The diameter of the tube also depends on the size of the field, and it will roughly be proportional to the length of the tube. If the width of the field is a span of 100˜150 meters, the suitable diameter of the tubes will be around 4-5 meters. Therefore, when we need the air beams to constitute the entire roof, we need the total numbers of the air beams to be about 40-60. If the length of the entire roof structure is 300 meters, the roof would need around 60 tubes.

3. The tubes are made of durable membrane. The tubes will become an arc shape after being inflated. Both ends of the tubes are cone shaped with materials of nylon-reinforced fiber glass plates and membrane. The shape of the cone, constituted by a pair of fiber glass plates, is changeable with the inflation and deflation process of the air beam (tube). At the end of the fiber glass plates, a steel block with sliding ditch and a steel wheel is installed. The sliding ditch and steel wheel will make the steel block, along with the air beam, be able to freely slide along a long sliding bar and a long C-channel track. The C-channel track and the sliding bar are parallely installed on the angle steel frame installed on the ground of the two sides of the field outside the stands, or on the existing ring shaped roof above the stands for spectators. These parts will be described separately later in detail. These parts make the air beams be able to freely slide along the track and sliding bar to realize the roof retracting and deploying.

4. All the air beams are equally divided into two groups. The air beams in each group will cover one half of the field. During the retracting process, the two groups of air beams will be deflated and be pulled to the opposite ends of the field separately and simultaneously. The total number of beams in each group is about 24˜30.

5. Between each pair of neighboring air beams, a membrane strip is installed on their top for rain proof, which is called the “rain strip”.

6. The air beams connected and paralleled together, but for each beam, there is a separate inflation and deflation vent. All of the vents are connected with two central inflator, one for each group of air beams. Each inflator supplies air through air ducts of a diameter of 15-25 centimeters.

7. When the air beams are deflated, they are folded, stacked, and neatly laid at both ends of the field and remain in an arc shape. In this case, the entire roof of the field is in an open status (retracted status).

8. When the weather is going to be bad, we need deploy the roof, we may adopt the opposite procedures:

i) turn on the two inflators at the two ends of the field, and

ii) inflate all the air beams successively (starts from the No. 1, and No. 2 beams),

iii) the inflated air beams will be gradually pushed to the center of the field, until

iv) the pair of the rain gutters, separately installed on the front one (No. 1 and No. 2 Beam) of each group of air beams, couple together. Then the deploy process is completed.

DETAILED DESCRIPTION OF INVENTION (I) Dynamics and Statics Analysis of Air Beam

The dynamics and statics analysis (force analysis) for the air beam will be the most fundamental and critical part of this invention.

We take a single air beam as the object to analyze the external forces exerted on the beam and internal forces (pressure force of air) inside the air beam.

All the external forces which exert on the air beam can be classified into the following categories:

The gravitational force of the membrane itself (including rain strip)

Wind force exerted on the top membrane of the air beam and rain strip

Weight of snow and hail accumulated on the air beam and rain strip

Supporting force from the supporting structure exerted on the two ends of the air beam

When we analyze the dynamics and statics for a single beam, we emphasize the forces from vertical direction and left-right direction. The forces from the neighboring air beams (air beams at the front-back direction) are neglected in consideration.

When we take a single air beam as the object of consideration, and its two ends are included in this object, the air pressure force inside the air beam and the tension force from the wind resistant rope will all be considered as internal forces.

All the forces are plotted in FIG. 1-4.

A. List of Symbols Including their Units

For simplicity, we listed the symbols and its units as follow:

L—Length of a single air beam, in m (meter)

D—Diameter of a single air beam, in m (meter)

σ—Thickness of membrane of air beam, in m (meter)

σ_(r)—Thickness of membrane of the rain strip, in m (meter)

ρ—Density of the membrane material, in T/m³

L′—Span of an air beam, in m (meter)

θ—Degree of the arc of air beam, in degree

R—Radius of the arc of air beam, in m (meter)

W—Total weight of a single air beam, in T (ton)

W_(r)—Total weight of a single piece of rain strip, in T (ton)

W_(s)—Total weight of snow accumulated on a single air beam, in T (ton)

W_(e)—Weight of each end portion of the air beam, in T (ton)

F_(w)—Total force of wind exerted on a single air beam (including rain strip), in T (ton)

φ—Angle of F_(w) downward from horizontal direction, in degree

N_(n)—Vertical component of the force from supporting frame to the North end of air beam, in T (ton)

N_(s)—Vertical component of the force from supporting frame to the South end of air beam, in T (ton)

F_(n)—Horizontal component of the force from supporting frame to the North end of air beam, in T (ton)

F_(s)—Horizontal component of the force from supporting frame to the South end of air beam, in T (ton)

N′_(n)—Vertical component of the force from the North end surface exerted to the air inside of the air beam, in T (ton)

N′_(s)—Vertical component of the force from the South end surface exerted to the air inside of the air beam, in T (ton)

F′_(n)—Horizontal component of the force from the North end surface exerted to the air inside of the air beam, in T (ton)

F′_(s)—Horizontal component of the force from the South end surface exerted to the air inside of the air beam, in T (ton)

φ_(n)—Total tension force of the four wind resistant ropes at the North side, in T (ton)

τ_(s)—Total tension force of the four wind resistant ropes at the South side, in T (ton)

M_(n)—Total force from the hard nylon reinforced plate at the North end exerted on the membrane at the North side, in T (ton)

M_(s)—Total force from the hard nylon inforced plate at the South end exerted on the membrane at the South side, in T (ton)

θ_(n)—Angle of the vector M_(n) downwards from the horizontal direction (see FIG. 4), in degree

θ_(s)—Angle of the vector M_(s) downwards from the horizontal direction (see FIG. 4), in degree

θ_(w)—Angle of the equivalent wind force downwards from the horizontal direction, in degree

θ_(r)—Angle of the resultant tension forces of the four ropes at the South side from the horizontal direction, in degree

_(r)—Distance of the action point of result rope pulling force at the South side to the South end point of the air beam, in m (meter)

_(w)—Distance of the action point of the result wind force of the entire air beam to the North end of the air beam (in case of North wind), in m (meter)

₁—Length of the membrane section taken in the statics analysis of stress, in m (meter)

σ_(s)—Weight of snow covered in each unit horizontal area, in T/m² (ton per square meter)

p—The amount of air pressure difference inside the air beam, which exceeds the atmospheric pressure, in T/m² (ton per square meter)

P—Tension stress of the membrane of the air beam, in T/m (ton per meter)

P₀—The maximum amount of tension stress, which the membrane material can endure, or the durability of the membrane, in T/m (ton per meter)

B. Individual Force Analysis:

All the external forces, which are exerted onto each single air beam, are as follows:

1. Weight of the membrane material of a single air beam. Suppose the total length of a single air beam is denoted by L, in unit meter, the diameter of the air beam is D in meter, the thickness of the membrane material of the air beam is in meter, the density of the membrane is in ton per cubic meter (which is equal to gram per cubic centimeter). The total weight, W, of the membrane material can be calculated as follow:

W=πDLσρ.  (1)

Here, is the density of the membrane material with a unit of T/m³. Usually, the density of the membrane material is about 0.85 T/m³. If we adopt D=4 m, L=100 m, σ=0.003 m (3 mm), and ρ=0.85 T/m³, then the total weight of the membrane material is 3.2 T.

2. Weight of two ends of the air beam. The weight of each end portion of the air beam, which includes the nylon reinforced hard plates, steel block and steel wheel, denoted as W_(e), is about 0.2 T. W_(e)=0.2 T

3. Weight of snow covered on a single air beam. This weight is only the total weight of the snow covered above each single air beam (it includes the snow on the rain strip above this air beam). The total weight of the snow on a single air beam, denoted as W_(s), can be calculated as follow:

W _(s) =DL′σ _(s).  (2)

From a simple geometric calculation, we can easily obtain the relationship between the span L′ and the length of an air beam L as follow:

L′=(3√3/2π)L or L′=0.827L  (3)

If L=100 m, then L′=82.7 m, D=4 m, σ_(s)=0.005 T/m². Then, W_(s)=1.65 T.

4. Weight of rain strip. If the width of the rain strip is S_(r), the thickness of the membrane material of the rain strip is denoted by σ_(r), and the density of the membrane material is ρ_(r), then the weight of a single rain strip will be:

W _(r) =S _(r) Lσ _(r)ρ_(r).  (4)

If L=100 m, S_(r)=1/2D=2 m, σ_(r)=0.002 m, and ρ_(r)=0.85 T/m³, then W_(r)=0.34 T.

5. Wind force exerted on a single air beam. Suppose the long axis of the stadium is from West to East, and the strong wind is from the North. The total force exerted onto a single air beam can be denoted as F_(w). The direction of F_(w) is denoted by θ_(w), which is the angle of the equivalent wind force downwards from the horizontal direction. The distance between the action point of the equivalent wind force and the North end supporting point of the air beam is denoted by

_(w). Usually,

_(w)=1/6 L. The detailed calculation will be in the following paragraphs:

6. Forces from the track exerted onto each end of the air beam. This force includes the force from the track exerted on the steel wheel at the end of the air beam, and the force from the sliding bar exerted on the sliding ditch. These two forces are combined together to form the total vertical component N_(n) and the total horizontal component F_(n) for the North end portion of the air beam. Similarly, we have the total vertical component N_(s) and the total horizontal component F_(s) for the South end portion of the air beam. Detailed calculation will be in the following paragraphs.

7. Forces from the end portion exerted on the air beam. For detailed calculation purposes, we need an imaginary cross-section at the ends of the air beam which separates the air beam and its end portion. This force includes two parts, as follows:

Part 1. Air pressure force at the cross-section. This force can be decomposed into the vertical component N_(n)′ and the horizontal component F_(n)′. Because the air pressure force must be perpendicular to the cross-section, N_(n)′+F_(n)′ will be 30° from the vertical direction if we adopt the air beam with an arc of 120°.

Part 2. The force from the end portion exerted on the membrane of the air beam. This force is denoted by M_(n). θ_(n) denotes the angle between M_(n) and the horizontal direction. This force does not include the air pressure force.

The above forces are for the North end portion of the air beam. Similarly, for the South end portion of the air beam, we have N_(s), F_(s), M_(s), and θ_(s).

C. Force Analysis for Single Air Beam in Case of Snow Covered but No Wind Considered:

We take one single air beam as the object in consideration. It may be covered with snow accumulation, but no wind is considered. Because heavy wind and snow accumulation will most likely not occur at the same time during one event at the stadium, so this assumption is reasonable. In the present calculation, we only consider the forces from the North and South direction and vertical direction. In other words, we consider the two-dimensional case (XZ plane), and neglect the forces in the Y direction. Because of the property of symmetry, the forces from the neighboring air beam exerted to the considered air beam do not play a critical role in the present calculation. Therefore, the present treatment is reasonable.

All the forces exerted on a single air beam are shown in FIG. 1. Because of the property of symmetry, all the forces W, W_(s), W_(r), and 2W_(e) can be concentrated at the centralized action point, the center of the air beam.

From the statics analysis, we have the following four equilibrium equations.

1. Equation for vertical components of forces.

W+2W _(e) +W _(s)+_(r) =N _(n) +N _(s).  (5)

2. Equation for horizontal components of forces.

F _(n) =F _(s).  (6)

3. Equation for torque in clockwise direction around the North end. We take the North end of the air beam as the reference point. The torque equation in clockwise direction

$\begin{matrix} {{\sum\limits_{i}{F_{i}d_{i}}} = 0} & (7) \end{matrix}$

can be written as follow:

(W+2W _(e) +W _(s) +W _(r))1/2L′−N _(s) L′=0.  (8)

4. Equation for torque in clockwise direction around the South end. We take the South end of the air beam as the reference point. The torque equation in clockwise direction can be written as follow:

−(W+2W _(e) +W _(s) +W _(r))1/2L′+N _(n) L′=0.  (9)

From Eqs. (8) and (9), we have

N _(n) =N _(s)=1/2(W+2W _(e) +W _(s) +W _(r)).  (10)

We may adjust the tension force of all the eight ropes such that

F _(n) =F _(s)=(1/√{square root over (3)})N _(n),  (11)

then N_(n)+F_(n) and N_(s)+F_(s) both are perpendicular to the cross-section of the air beam. So, they are in the same direction of the pressure force of air at the cross-section. Therefore, the membrane tension force at the joint at the end portion can be 0 when we properly adjust the air pressure by the inflating machine and the tension of the ropes.

As calculated above, when we have W=3.2 T, W_(s)=1.65 T, W_(e)=0.2 T, and W_(r)=0.34 T, from Eqs. (6), (8), and (9), we have

N _(n) =N _(s)=1/2(W+2W _(e) +W _(s) W _(r))=2.8 T,  (12)

F _(n) =F _(s)=(1/√{square root over (3)})N _(n)=1.62 T,  (13)

|N _(n) +F _(n) |=|N _(s) +F _(s)|=(N _(n) ² +F _(n) ²)^(1/2)=π(½D)² p=3.24 T,  (14)

where p is the air pressure force in addition to the atmospheric pressure. From Eq. (14), we obtain

p=3.24/π(½D)² , p=0.258 T/m², for D=4 m  (15)

D. Membrane Tension Stress Calculations

Suppose we take a section of the membrane of length

₁, take a half of it, and examine the stress on its cross-section (see FIG. 31). The tension stress of the membrane is denoted by P. The amount of air pressure difference inside the air beam, which exceeds the atmospheric pressure, is denoted by p.

Obviously, the equilibrium equation in the direction shown will be:

₁ Dp=2

₁ P,  (16)

from which we obtain

P=1/2Dp.  (17)

As calculated above, p=0.258 T/m² for D=4 m, we obtain P=0.516 T/m, which equals P=5.16 kg/cm. Most membranes with this thickness, 3 mm, will have stronger durability than this amount. So, our air beam will work with no trouble.

From Eq. (14), we have

(N _(n) ² F _(n) ²)=^(1/2)=π(½D)² p, or p=(N _(n) ² F _(n) ²)^(1/2)/π(½D)².  (18)

Substituting Eq. (18) into Eq. (17), we have

P=(1/πD)(N _(n) ² +F _(n) ²)^(1/2).  (19)

From Eqs. (10)-(13), we have

N _(n) ,N _(s) ,F _(n) ,F _(s),(N _(n) ² +F _(n) ²)^(1/2),(N _(s) ² +F _(s) ²)^(1/2) ∝W+2W _(e) +W _(r) +W _(s).  (20)

From Eqs. (1)-(4) and related analysis, we have

W+2W _(e) +W _(r) +W _(s) ∝D.  (21)

From Eqs (20) and (21), we obtain

N _(n) ,N _(s) ,F _(n) ,F _(s),(N _(n) ² +F _(n) ²)^(1/2),(N _(s) ² +F _(s) ²)^(1/2) ∝D.  (22)

From Eqs. (19) and (22), we conclude that P is independent of D. In other words, when we increase the diameter of the air beam, it will not increase the membrane tension in the present case.

E. Case of Strong Wind Exists but No Snow Accumulation

Suppose the long axis of the stadium is in the West-East direction, and the strong wind is from the North. The total (resultant) wind force exerted on a single air beam is denoted by F_(w). The action point of F_(w) is located at a point, which has distance

_(t), from the supporting point at the North end of the air beam (see FIG. 2).

By the fluid mechanics analysis, we can accurately obtain

_(w)=(⅙) L, and the direction of F_(w) is accurately pointed to the South end point of the air beam. The angle between the wind direction and the horizontal direction is denoted by θ_(w). From a simple geometric calculation, we obtained that, θ_(w)=10°, when we adopt the air beam with an arc of 120°. Moreover, when the wind is from the North, the total resultant tension force of the four ropes at the South side of the air beam, which is denoted by τ_(s). The angle between τ_(s) and the horizontal direction is denoted by θ_(r).

From the dynamics and statics force analysis, we conclude that:

i) τ_(s) will be proportional to the wind force F_(w),

ii) the distance between the action point of τ_(s) and the South end of the air beam, denoted as

_(r), is also close to ⅙ L for the air beam of arc 120°,

iii) the angle between τ_(s) and the horizontal direction, denoted by θ_(r), is also 10° for the air beam of arc 120°. The direction of τ_(s) is definitely pointed to the North end points of the air beam.

For the strong wind from the North direction, the resultant tension force of the four ropes at the North end, denoted by τ_(n), will not increase, but even decrease to 0. In other words, τ_(n) will be neglected.

When we choose the object to write the force and torque equations, we take the following two choices.

Choice 1. Take the entire air beam, including the two ends, as our object. In this case, all the external forces exerted on the object will be W, W_(r), W_(e), F_(w), N_(n), F_(n), N_(s), F_(s). Here, N_(n), F_(n), N_(s), F_(s) are the vertical components and the horizontal components of the track exerted on the North end and South end of the air beam, which includes

i) the force from the track (c-channel) exerted on the steel wheel

ii) the force from the sliding bar exerted on the sliding ditch of the block So, the forces N_(n), F_(n), N_(s), F_(s) all are the resultant force of the above two items.

In this choice of object, τ_(s) and τ_(n) will be the internal forces which do not appear in the following force equilibrium equations.

i) Force equation for vertical direction component is as follow:

W+2W _(e) +W _(r) +F _(w) sin θ_(w) =N _(n) +N _(s).  (23)

ii) Force equation for horizontal direction component is as follow:

F _(w) cos θ_(w) +F _(n) =F _(s).  (24)

Here, N_(n), F_(n), N, F_(s) are algebraic quantities according to the positive directions shown in FIG. 2.

iii) Torque equation for the reference point at the North end of air beam in clockwise direction:

(W+2W _(e) +W _(r))1/2L′+F _(w) L′ sin θ_(w) =N _(s) L′  (25)

iv) Torque equation for the reference point at the South end of air beam in clockwise direction:

−(W+2W _(e) +W _(r))1/2L′+N _(n) L′=0  (26)

After solving the equations (23)-(26), we obtain the solution as follow:

N _(n)=1/2(W+2W _(e) +W _(r))  (27)

N _(s)=1/2(W+2W _(e) +W _(r))+F _(w) sin θ_(w)  (28)

To obtain a better result in minimizing the tension stress of the membrane, we need more detailed dynamics and statics analysis in the following paragraphs.

Choice 2. Take the single air beam without their two end portions as the object.

We take one single air beam as the object of consideration, and set up two imaginary cross-sections at its two ends between the main membrane and its two end portions, as shown in FIG. 4.

In this case, all the external forces exerted on the object are: W, W_(r), F_(w), N_(n)′, F_(n)′, N_(s)′, F_(s)′, M_(n), M_(s), τ_(n), τ_(s). Here, M_(n) and M_(s) are forces from the end portions (the nylon reinforced hard plates and its membranes filled in) exerted onto the main membrane of the air beam at the North and South, respectively. The directions of the M_(n) and M_(s) are determined by θ_(n) and θ_(s), respectively (see FIG. 4). W_(e) is not included in the external forces.

For the present case, the four force and torque equations are as follows:

i) Force equation for vertical components:

W+W _(r) +F _(w) sin θ_(w)+τ_(s) sin θ_(r) +M _(n) sin θ_(n) +M _(s) sin θ_(s) =N _(n) ′+N _(s)′.  (29)

Here, F_(w), M_(n), and M_(s) are the modulus of the vectors F_(w), M_(n), and M_(s), respectively. So, they are always positive.

ii) Force equation for horizontal components:

F _(w) cos θ_(w)−τ_(s) cos θ_(r) +F _(n) ′−F _(s) ′−M _(n) cos θ_(n) −M _(s) cos θ_(s)=0  (30)

iii) Torque equation with North end point of the air beam as reference, in clockwise direction:

½L′(W+W _(r))+F _(w) L′ sin θ_(w) +M _(s) L′ sin θ_(s) =N _(s) ′L′  (31)

iv) Torque equation with South end point of the air beam as reference, in clockwise direction:

−1/2L′(W+W _(r))+N _(n) ′L′−τ _(s) L′ sin θ_(r) −M _(n) L′ sin θ_(n)=0  (32)

Solving for above four Eqs. (29)-(32), we can obtain the solutions regarding the tension stress of the membrane in strong wind cases as follows:

From Eqs. (31) and (32), we obtain:

N _(s)′=1/2(W+W _(r))+F _(w) sin θ_(w) +M _(s) sin θ_(s),  (33)

N _(n)′=(W+W _(r))+τ_(s) sin θ_(r) +M _(n) sin θ_(n).  (34)

We noticed that these two solutions just coincide with Eq. (29).

Because N_(n)′+F_(n)′ is the total air pressure force, which must be perpendicular to the interface, that is 30° incline from the horizontal, we have

F _(n) ′/N _(n)′=tan 30°, F _(s) ′/N _(s)′=tan 30°,  (35)

i.e.

F _(n) ′=N _(n)′ tan 30°=0.577N _(n) ′, F _(s) ′=N _(s)′ tan 30°=0.577N _(s)′.  (36)

Substituting Eq. (36) into (30), we have

M _(n) cos θ_(n) +M _(s) cos θ_(s)=0.577(N _(n) ′−N _(s)′)+F _(w) cos θ_(w)−τ_(s) cos θ_(r).  (37)

From Eqs. (33) and (34), we obtain

N _(n) ′−N _(s)′=τ_(s) sin θ_(r) −F _(w) sin θ_(w) +M _(n) sin θ_(n) −M _(s) sin θ_(s).  (38)

Because, in one tube, the air pressure must be equal everywhere, we have

|N _(n) ′+F _(n) ′|=+|N _(s) ′+F _(s)′|=2F _(n)′=2F _(s)′  (39)

and

N _(n) ′=N _(s) ′, F _(n) ′=F _(s)′.  (40)

Here, we have to remind that, N_(n)′, N_(s)′, F_(n)′, and F_(s)′ all are algebraic quantities and their positive directions are indicated in FIG. 4.

Substituting Eq. (40) into Eqs. (37) and (38), we obtain the relationship equations as follows:

M _(n) cos θ_(n) +M _(s) cos θ_(s) =F _(w) cos θ_(w)−τ_(s) cos θ_(r)  (41)

and

M _(n) sin θ_(n) −M _(s) sin θ_(s) =F _(w) sin θ_(w)−τ_(s) sin θ_(r)  (42)

From FIG. 4), we can easily see the range for θ_(n) and θ_(s) as follows:

θ_(n)∈[−30°,+150°], θ_(s)∈[+30°,+210°]  (43)

Solving for Eqs. (41) and (42) and combining with the above Eqs. (33) and (34), we obtain the relationship equations as follows:

M _(s) sin θ_(s) =N _(s)′−1/2(W+W _(r))−F _(w) sin θ_(w),  (44)

M _(n) sin θ_(n) =N _(n)′−1/2(W+W _(r))−τ_(s) sin σ_(r).  (45)

F. Forces Analysis Between Neighboring Air Beams

1. During deploying process: When the beams are in the deployment process, there are compressing forces between each pair of neighboring beams. The amount of these forces depends on the location of the pairs and the pressure of air inside the beams. The biggest compressing force will be between the beams number 47 and 45 and between the beams number 48 and 46 at the beginning of inflation. The compressing force between the pair of beams number 47 and 45 will be the total weight of the membrane of the 23 beams above it. The compressing force between the pair of beams No. 45 and 43 will be the weight of 22 beams above it . . . and so on. These forces will not cause any problem during the operation.

For the other group of beams on the other side, the condition of the compressing forces will be similar.

2. During retracting process: When the beams are in the retraction process, there are pulling forces between each pair of neighboring beams. The pulling forces between each pair of neighboring air beams are replaced by the tension force of the rain strip above the beams and the pulling force by the pulling chain at the bottom portion of the beams. Because the pulling force by the pulling chain will play the main role, the tension force in the rain strips will not cause any problem.

Important Conclusions and Results

From the above Eqs. (44) and (45), we can obtain the following significant and important results.

i) during heavy North wind, the rope tension force at the South side will increase proportionally, and the rope tension forces at the North side remain zero.

ii) the force from the South end portion of the air beam exerted onto the main membrane of the air beam will be closely related to the total wind force.

iii) the force from the North end portion of the air beam exerted onto the main membrane of the air beam will be closely related to the rope tension force at the South side.

iv) during the heavy wind, F_(w) and τ_(s) are increased. In order to reduce the M_(s) and M_(n) (the forces from the end portion exerted onto the main membrane), we may turn on the inflator to increase the pressure inside the air beam, which will increase |N_(n)′+F_(n)′| and |N_(s)′+F_(s)′|, and equivalently increase N_(n)′ and N_(s)′.

v) during the heavy wind, we may adopt two methods:

a. to adjust the air pressure inside the air beam by turning the inflator on and carefully controlling the air pressure.

b. to adjust the rope tension force by changing the length of ropes on both sides. By the above careful adjustments, we may finally reach the result that M_(s) and M_(n) equal to zero or neglectably small. When M_(s), and M_(n) become zero or neglectably small, Eqs. (33) and (34) can be simplified to

N _(s)′=1/2(W+W _(r))+F _(w) sin θ_(w),  (46)

N _(n)′=1/2(W+W _(r))+τ_(s) sin θ_(r).  (47)

vi) no matter in the case of snow accumulation or in the case of strong wind, we may adopt the two adjustments: a. adjustment of the air pressure inside the air beam. b. adjustment of the rope tension, to finally make the entire roof force become a pure air pressure onto the two ends portion. In this way, we finally made the entire roof force (including the weight of material, snow, and wind) become the air pressure onto the both ends portion, and minimized the various stresses of the solid material.

Numerical Calculations

In the case of strong wind existing and no snow accumulation, one may also adopt the following numerical data: D=4 m, L=100 m, σ=0.003 m, σ_(r)=0.02 m, ρ=0.85 T/m³,

_(w)=1/6 L=16.67 m,

_(r)=1/6 L=16.67 m, θ_(w)=10°, θ_(r)=10°. From the above numerical results, we have W=3.2 T, W_(r)=0.34 T. From the fluid dynamics analysis, when the wind is between level 8 and level 9, F_(w) will be close to 2˜3 ton. We take F_(w)=3 T. From Eq. (46), we obtain N_(s)′=2.291 T, |N_(s)′+F_(s)′|=2.645 T. From (½ D)² p=|N_(s)′+F_(s)′|, we obtain p=0.21 T/m², P=1/2 D p=0.42 T/m (or 4.2 kg/cm).

(II) Force Analysis in Pulling Chains

The winding machine and the pulling chain work only in the process of retraction and deflation. They do not need to work during the process of deployment and inflation.

The maximum pulling force of a chain occurs at the beginning of retraction. This force relies on the total number of beams in each group and the total force from the end portion exerted on the sliding track, which is (N_(s) ²+F_(s) ²)^(1/2). The operation of pulling by the chain is just similar to the operation of a locomotive pulling freight carriers of a train. The total pulling force from the locomotive is about 1/1000 (or less) of the total weight of the train. In our case, the total force from the ends of beams exerted onto the sliding track is 24 (N_(s) ²+F_(s) ²)^(1/2)=24·2.645 T. Therefore, the maximum pulling force will be

F _(p)=( 1/1000)·24·2.645=0.0635 T=63 kg.

This amount of pulling force can be easily handled by a regular chain and winding machine. Because the friction coefficient of our system is much larger than the train system, the coefficient 1/1000 should be replaced by a bigger amount, say, 1/400, then the maximum pulling force F_(p) will be 158 kg. It still can be easily handled.

(III) Structure of the Entire Roof Part 1. Parts of Air Beam:

The air beams are the central and most important part of the entire roof. They consist of the following portions:

1. Arc shaped membrane tube. The length of the tube will be easily obtained from the length of the span of the roof opening. The length of span can be about 100˜150 meters usually, which depends on the opening of the existing ring roof above stands for spectators. The suitable angle of the arc of the beam is about 120° (see FIG. 1). The diameter of the tube, when inflated, is about 3.5˜5 meters, which depends on the length of the tube. The material of the tube will adopt the most durable material (with strongest tension intensity), for instance, the nylon reinforced polyethylene. The thickness of the membrane material depends on the diameter of the tube. Usually, the thickness is 3˜6 mm.

2. Arc shaped solid reinforced fiberglass strip: In order to strengthen the intensity of the membrane tube, the joint of the membrane material will be at the lower side and connected by a pair of strips, which are made of hard fiberglass. The strips are in an arch shape, which are in the same shape and same length as the inner side of the air beam. The thickness of the strips are about 0.5˜1 cm, and the width of the strips are about 7˜10 cm. There are 8 holes along the strips, which are for the nylon rope connection (see FIG. 15).

3. Structure of the two ends of each air beam. At the two ends of each air beam, the structure is sophisticated. The end portion consists of the following parts:

i) Hard fiberglass plates. There are two fiberglass plates with shape as FIG. 6. The two plates are connected with a hinge with opening angle from 0° to 60°. The plates are in isosceles triangle shape with a base length of 2˜4 meters, which should match the size of the air beam. The height of the triangular plates are 3˜6 meters. The angles of the isosceles triangles are 50°, 65°, and 65°

ii) Strongly reinforced soft fiberglass membrane material. Between the two fiberglass plates, there is reinforced by two pieces of soft fiberglass material with isosceles triangle shape and tightly connected to the two plates. The size of the soft material will be the same as the two plates. When the air beam is deflating, the two plates will move towards an angle of 0° position. In the same time, the soft fiberglass material will be slowly folding towards the inside direction.

iii) Elaborately designed sliding structure. This block is made of hard fiberglass or steel. The block is able to slide along the track. The two fiberglass plates are connected to the upper end of the block with hinges, which allow the two fiberglass plates to open and close symmetrically and simultaneously. The spherical hinge is connected to the lower end of the block.

iv) Connection of cone shaped ends with the air beam. The two hard fiberglass plates and the two pieces of soft fiberglass material mentioned above form a cone with 4 sides. The perimeter of the base of the cone should be exactly equal to the circumference of the opening of the air beam. The ends of the air beam and the cone openings will be tightly connected by strong glue.

v) Vent valve with motor for inflation and deflation. There is one vent valve installed at one of the two ends of the air beam, which is installed on the hard fiberglass plates.

4. Strengthening wind resistant ropes. On each air beam, there are 8 nylon ropes installed on certain positions, which is for strengthening the beam in especially strong wind conditions. Ropes are connected through the holes on the strips and steel blocks.

Part 2. Rain Strip (Membrane) Between Beams:

In order to protect from rain, snow, hail, and wind, there is membrane material in a wide strip shape installed above all gaps between two neighboring beams, except between beam 1 and beam 2. Each wide strip will have the same length as the air beam and width about 2˜3 meters or ½ of the diameter of the air beam. The material of the strips can be the same transparent or translucent material as the tubes, but the tension intensity can be lower. The strips are strongly glued with the tubes.

Part 3. Inflating Machine and Air Duct:

The inflating machine is the central air supplier of the entire roof. It must be reliable. If we need to deploy the roof within 20 minutes, the inflating machine should be able to supply 10,000˜20,000 cubic meters of air per minute. The maximum pressure of the air will depend on the durability of the membrane material.

When retracting the roof, to speed up the deflating process, we will use the same inflating machine to draw the air in the opposite flow direction.

Air ducts are connected between the inflating machine and the vent valves of each air beam. When the roof is inflating, the air flows from the inflating machine to all the air beams. Because we need the air beams to be inflated one after another, each air beam has its own air duct. As the air beams slide along the tracks, the air ducts should be moved and collected through elaborate design.

Part 4. Steel Block:

A steel block is installed at the end of the air beam, which is connected to the end by using a hinge. When the two nylon reinforced hard plates of the end turn between 0° and 50° during the inflation and deflation of the air beam, the block will remain not turning. At the lower end of the block, a steel wheel is installed (see FIG. 16).

Part 5. Steel Wheel and Sliding Ditch:

There is one steel wheel installed at each end of the steel block. The steel block is elaborately designed and installed onto the connection hinge of the two hard fiberglass plates. The steel wheel is connected to the block with a ball bearing. On the opposite side of the block, there is a sliding ditch with cylindrical bearing, which makes the block be able to slide along the sliding bar. The steel wheel will be able to freely slide along the track and sliding bar. Because all air beams are divided into 2 groups, one on each half of the field, so there are 4 groups of steel wheels.

Part 6. Sliding Track and Sliding Bar:

1. Sliding Track. The tracks are installed on both sides of the field. The tracks are made of steel c channels and two rails. If the stadium has stands without roof structure covered, the two tracks will be installed behind the stands on the specially designed supporting frame on the base of the stadium. If the stadium has stands covered with a ring shaped roof above the stands for the audience, the two tracks will be installed on the edge of the roof above the steel structure. The tracks are parallel and straight, but both ends will be curved down in a shape of quarter circle (see FIG. 17) in order to make all the air beams be positioned correctly in both deploying and retracting status. All steel wheels can be slided along the tracks. The height of the track from the ground (in case of no roof structure above the audience, we call this case 1) or from the existing roof structure (in case of existing roof structure above the audience, we call this case 2) depends on the height of the pile of deflated roofs, which depends on the number of air beams and thickness of material. In case 1, the height of the track will be around 2˜4 meter built from the ground. In case 2, the height of the track will be 2˜3 meters from the existing ring roof structure.

2. Sliding Bar. The sliding bar is installed on the angle steel frame, together with the track, but the bar and the track are on the opposite side of the steel block. The sliding bar is parallel with the track. Both the track and bar have the same length and have accurate relative positions. The sliding bar is well lubricated and made of stainless steel. Just like the track, there are four sections of sliding bars for the entire stadium. (see FIGS. 19, 20, 21)

3. Supporting Structure of Track.

i) If the track is built outside of the stands (in case of no existing ring roof above the stands, that is, case 1), it will be 2˜4 meter above the ground. In this case, the steel frame, which supports the track from the ground, will be triangle shaped and made of steel angle of 6˜10 cm. We need a number of steel frames to support the track spaced with a distance of about 3˜5 meter.

ii) If the track is built above the ring roof (in case of having an existing ring roof above the stands, that is case 2), the supporting frame will be built from the steel structure of the existing ring roof. The material and shape of the supporting frame can be the same as above, but the size can be smaller, that is, 2˜3 meter tall. (see FIGS. 19, 20, 21)

Part 7. Pulling Chain:

Chains are used to pull the steel blocks sliding along the track and sliding bar. Because the blocks are connected with the end portion of the air beam, the pulling chain will cause the entire roof to move. Each chain will pull one group of blocks. Because there are four groups of end portions of the air beams, there are four chains to pull these four groups of end portions of air beams. One end of each chain is connected to the steel block of the ending air beam, that is air beam 1 and air beam 2. The other end of the chain is connected to the winding machine. The length of the chain is longer than the distance between the winding machine and the end portion of air beam 1 (or air beam 2) at the deployed position.

Part 8. Winding Machine:

There are four winding machines installed at the both ends of each track. Each winding machine is for pulling one of the chains. In the case of retracting the roof, turn on all four winding machines to pull the four chains simultaneously. The pulling speed will be the same as the speed of deflation and adjusted to synchronize. When all the air beams are deflated and folded in piles, the winding machines will be turned off. In the case of deploying the roof, the air beams will be moved by the pushing force between the air beams during the inflation process, which will push the steel block to slide along the track. As a result, the chains will be automatically pulled out from the winding machine. Therefore, the winding machine does not need to be powered on during deployment.

Part 9. Vent Valve:

There is one valve installed on each air beam. This valve can be in open status or closed status. When the air beam is inflating, the valve is in open status (open to inside of the air beam). When the inflation is finished, the valve will be pushed by a small electric motor to close status (closed from inside of the air beam in order to have air pressure to seal the valve). When deflating the air beam, the electric motor will push the valve to open status. When the deflation is finished, the valve should be pulled back to closed status by the same motor.

Part 10. Central Control Switch Board:

All the parts, which need power to operate, are controlled by a central switch board through elaborately designed wiring. Adding a proper program, we may easily realize the completely automatic control, even automatically operating completely by the weather forecast.

(IV) Operating Procedures

A. Deploying Procedures: When the weather condition is becoming undesirable and the roof is needed, we will adopt the following procedures to deploy the roof:

1. Turn on the two inflators located on both sides of the field and send air through the air ducts and valves to the top beam of the two piles of deflated beams on both ends of the field simultaneously. We call these two beams “beam number 1” and “beam number 2”. During the inflation of beam number 1 and beam number 2, the two end structures of the beams are moved for some distance along the track.

2. After beam 1 and beam 2 are fully inflated, the valves for beam 1 and beam 2 are automatically closed and sealed. Then, the inflator starts to send air to beam 3 and beam 4 on both ends. During the inflation of beam 3 and beam 4, the four end structures of the beams are moved for some distance along the track.

3. After beam 3 and beam 4 are fully inflated, the valves for beam 3 and beam 4 are automatically closed and sealed. Then, the inflator starts to send air to beam 5 and beam 6 . . . and so on until all air beams are inflated.

4. After all the beams are inflated, Beam 1 and beam 2 will be touching each other, the inflator is stopped, and the entire roof is completely closed. In the same time, the specially designed rain gutters, which are mounted on beam 1 and beam 2, will be in the coupled position (FIG. 15).

B. Retracting Procedures: When the undesirable weather is over, the roof can be retracted. We will adopt the following procedure for retracting the roof:

1. Turn on the motor to open the valve for the last air beams on both ends, say beam 47 and beam 48 (supposedly we use 48 beams in total). In the meantime, turn on the inflators inversely to let the air out from beam 47 and beam 48. At the same time, also turn on the winding machines for winding the chains to pull the end structures backwards. During the deflation of air beam 47 and beam 48, the end structures of all the beams will move back for some distance along the track.

2. After beam 47 and beam 48 are fully deflated, the valve for beam 45 and beam 46 will be opened in the same way, and the inflator will begin to deflate beam 45 and beam 56 . . . and so on until all the beams are deflated and moved to the original spots.

3. After the beams are completely deflated, the entire roof will be in the retracted status, and the deflated beams will be neatly folded and stacked in order on the two ends of the field.

The entire process of deploying and retracting the roof will be inaudible for the audience and players. There will not be any disturbance, just like the clouds moving in the sky.

(V) Status of Deployed and Retracted Roof A. Status of Deployed Roof:

1. Air condition. When the roof is fully deployed and in closed condition, the air inside the field is in circulation as usual for the audience and players. The air pressure is at regular atmospheric pressure. There is no additional pressure to the audience as in common domes with bubble roofs. There is no effect on air quality inside the stadium, and the temperature can be adjusted freely as usual.

2. Illumination. The material of the membrane is transparent or translucent. So, beyond daylight, additional lighting for the field may not be needed during the daytime. If additional electric illumination is desired, the lights can be installed along the edge of the roof opening, under the tracks.

3. Undesirable Weather Protection. The roof can protect the field from rain, snow, hail, wind, and any other undesirable outdoor conditions. The wind speed or Beaufort level of the wind which the roof can stand, depends on the durability of the membrane material of the air beams. For our design, the regular durable membrane, such as 0.5 cm thick nylon membrane, could withstand wind of 9th level or wind with a speed of 85 km/h. The same kind of membrane will be able to withstand rain, snow, and hail of ordinary levels.

4. Odor. Due to the odorless membrane material that is adopted in the present design, there is no odor or any uncomfortable smell to the audience.

B. Status of Retracted Roof:

When the roof is retracted, the entire field will return to the status without a roof. The two piles of roof material are located at the two ends of the field behind the stands (in case of no roof covering stands) or on the existing ring roof structure (in case of stands covered by the existing ring roof). No matter in which case, the two piles of roof material are out of sight for the audience and players.

(VI) Measures of Wind Resistance

Because the main material of our roof is membrane, the wind resistance is a critical task for us. We have adopted the following measures to ensure that wind resistance is attained.

1. We have adopted the arc degree of the air beams, 120°, instead of 180° or 150°. This greatly reduces the wind pressure.

2. We have installed nylon ropes for each air beam to strengthen the ability of wind resistance.

3. We have adopted arc shaped hard fiberglass strips, installed on the inner side of the entire air beam (FIG. 15, FIG. 27, and FIG. 28). This strengthens the wind resistance.

4. We have adopted isosceles triangle shaped hard fiberglass plates (FIG. 5 and FIG. 6) for both two ends of air beams. This strengthens the wind resistance.

5. We have adopted end structures on each air beam, which are mounted on the sliding track in a way that it can slide smoothly and is non-separable. Therefore, it is impossible for the beams to dislocate no matter how strong the wind is.

(VII) Consideration of Special Undesirable Cases

1. Extraordinarily Strong Storm.

If there is extremely strong wind or storm, which is beyond usual levels, we may carefully follow the emergency weather forecast and emergency guide, and adopt the following emergency measures.

i) Stop the performance in the field.

ii) Direct the audience and players to take shelter at the area around the side of the stands.

iii) Retract the roof to avoid damage.

2. In Case of Fire.

Because all the membrane material and other materials of our roof are non-inflammable, it is impossible to cause fire to the entire roof. So, it is safe in the aspect of fire.

3. Earthquake.

Because of the entire structure of our roof, as described as above, it will be able to withstand almost all levels of earthquakes. The situation of “collapse” will not exist.

4. Power Outage.

In case of power outage, no matter in the deployed status, retracted status, or during the deploying and retracting process, it will not cause any danger to the audience and players.

5. In Case of Criminal Attacks and Shootings.

In case of criminal attacks and shootings, which cause air beams to be broken and deflated, as long as some air beams are still inflated, the entire roof will not fall down. Even if the entire roof falls down, its deflation will be a very slow process, because damage by bullets or other weapons would create holes, which are much smaller than our vent valve. Therefore, the falling of the roof will be very slow and last many hours. So, even if the roof is attacked and completely (every air beam) broken, it would not cause any injury to the people.

(VIII) Economic and Cost Analysis

Part 1: Materials and Parts Costs

For our roof, the main material is membrane. Therefore, the material costs will be mainly for the membrane. Membrane cost will be 70% of the entire material costs of the roof. The material and parts costs are roughly estimated as follow:

Name of material and parts Amount needed Unit price Total price 1. Membrane (3 mm thick) 97,200 m²  $5.60/m² $544,320.00 2. Membrane (2 mm thick) 15,600 m²  $3.50/m² $54,600.00 3. Soft fiberglass (5 mm thick) 810 m² $10.00/m²  $8,100.00 4. Hard fiberglass (10 mm thick) 400 m² $28.00/m²  $11,200.00 5. Stainless steel 9.4 T $1900.00/T     $17,860.00 6. Steel angle (5″ 5″ 0.5″) 810 m (29,160″) $2.60/in $75,816.00 7. Nylon rope (½″) 2,430 ft  $169.00/500 ft $821.34 8. Chain (stainless steel) 2,000 ft $200.00/50 ft $8,000.00 9. Flexible air duct (18″ diameter) 6,480 ft $280.00/25 ft $72,576.00 10. Inflating machine (10,000 m³/min) 2 $5000.00/each  $10,000.00 11. Chain winding pulley with motor 4 $1000.00/each  $4000.00 12. Vent valve with motor 108  $200.00/each $21,600.00 13. Central control switch board 1 $800.00/each $800.00 14. Wiring $1,000.00 Total cost of material and parts: $830,693.34

Part 2: Labor Costs

Titles Hours needed Wage/hour Total wage 1. General engineers and stadium specialists 100 $160/hr $15,000 2. Membrane engineers and specialists 200 $140/hr $30,000 3. Mechanical engineers and specialists 150 $150/hr $22,500 4. Electrical and electronical engineers 120 $150/hr $18,000 5. Membrane workers 1,920  $80/hr $153,600 6. Mechanical workers 2,560  $80/hr $204,800 7. Electrical and electronical workers 640  $90/hr $57,600 8. Other labors 200  $40/hr $8,000 Total labor costs: $509,500

Part 3: Maintenance Costs

The above two kinds of costs are for the construction of the roof. After the construction is finished, in the constant operations, the maintenance costs are comparatively low in comparison with other kinds of roofs.

In the ordinary operation, say 5 times per week, the maintenance costs will be as follows:

Type of work Hours needed/month Cost/hour Total cost/month 1. Cleaning 80 $35/hr $2,800 2. Snow removing 32 $35/hr $1,120 3. Check of membrane 75 $40/hr $3,000 leaking and repair 4. Check of other parts, 50 $40/hr $2,000 equipments, and lubrication Total maintenance costs: $8,920 per month

Part 4: Costs in Operation

Because our roof structure is absolutely safe in operation, the insurance cost can be almost zero. We adopted simple structures and easy operations, so the power consumption will be very low and can be neglected in operation.

Part 5: Marketing Assessment

The present invention suggests a very safe, usable, convenient, and low-cost roof structure for all kinds of fields. Not only for stadiums, but also for soccer, track and field, football, baseball, swimming, or any types of fields for performing arts. Also, it can be used for huge greenhouses for elaborate planting, nursing, and farming, which require big space for machine operating inside without obstacles, and require the roof to be transparent and constantly openable. Therefore, our invention has great marketing potential.

(IX) Comparison with Other Types of Roofs

The presently existing constructions and patents for retractable roofs can be categorized as follows:

1. Sliding blocks (panels). This type of retractable roof usually consists of two or more pieces of huge blocks or panels. The size of one block will reach around a hundred meters. The material includes large amounts of steel beams, fiberglass, and other solid and heavy materials. The total weight of one block reaches many thousand tons. The huge blocks slide along tracks to realize deploying and retracting.

2. Cable supported roofs. Some retracting roofs consist of very high and huge towers outside of the stadium, or in the center of the stadium. Many long, steel cables connect the retractable roofs to the towers to realize deploying and retracting.

3. Umbrella shaped membrane suspended by cables. Another type of retracting roof is made of membrane suspended by steel cables which is connected to the edge of the steel frame of the existing ring roof above the stands. The membrane can be pulled to the center of the stadium to realize the retraction process. The weight of the supporting cables reaches over 1,000 tons.

4. Sliding roof with steel frames and inflated membrane material. In Japan, there is a stadium roof with sliding blocks made of steel frames and inflated membrane material. Because the inflated material is used, the total weight can be reduced in comparison with previously noted sliding blocks. But, it is still heavy and difficult to operate. Since 2015, it has not been closed until now due to the high cost of operation and maintenance.

In comparison with the above 4 kinds of retractable roofs, our roofs have the following outstanding strong points:

1. Better safety. Because soft membrane is the main material of our roof, in any case, our roof is more safe than any one of the above roofs. Even though the 4th type of roof listed above is relatively safer than the first three kinds, our roof is still much safer than it.

2. Easier construction and lower cost. In comparison with all four kinds of roofs, the construction of our roof is much easier, and the cost for material and labor are much lower.

3. Easier operation. The deploying and retracting process for our roof are simply achieved by a switch board operated by one person.

4. Easier maintenance. Due to the lightweight, simple structure and material, this makes our maintenance much easier than the other four kinds.

5. Better visual and environmental conditions for audience and players. Our roof has better transparent and translucent ability and absolutely no noise during operation.

6. Stronger ability to resist emergency and undesirable situations. 

1-3. (canceled) 4: An Air Beam, comprising: an Air Body, shaped as a cylinder of certain length and certain diameter, and bended as a circular arc with certain arc degree; a Cylindrical Shaped Soft Membrane Tube (Air Tube) of the same size and same shape (when fully inflated) as the said Air Body, made of durable membrane material (for example, nylon reinforced fabric), with the function of confining the said Air Body and separating the Air Body with the outside atmosphere; an End-Structures installed at each end of the said air tube, made of both soft membrane and solid material (such as nylon reinforced fiberglass), which has three functions: A. sealing the air, B. supporting the air beam and the load (if any) on the air beam, C. sliding (moving) along a track to change the location of said air beam; is initiated and established by inventor Keh C. Liu (Jun. 2, 2006 filed in the National Patent Office of China, Beijing, File No. 200610083730.8, titled “Air Beam and Inflatable, Retractable Mobile Roof for Large Spaces”). 5: A Method of Math and Phys Derivation for the physical characteristics of a said single Air Beam, and of the entire Retractable Air Beam Roof of large size, under various weather conditions, comprising: using Rigorous mathematics derivation and dynamics and statics analyses; to finding the relations (calculation formulas) between the length, diameter, span, arc degree of the said Air Beam, the wind speed, wind direction, snow accumulation thickness, the air pressure inside the air tube, the tension stress in the membrane, especially at the end portion of the air tube, and the durability of the membrane material; which forming the foundation and basis for properly, correctly and quantitatively designing the Retractable Air Beam Roof; and resulting that, a properly designed retractable air beam roof with regular membrane durability will be able to work for large stadium of over one hundred meter span and over 180 meter length and for any regularly severe weather conditions. is initiated and established by inventor Keh C. Liu (Jun. 2, 2006 filed at National Patent Office of China, Beijing, File No. 200610083730.8, titled “Air Beam and Inflatable, Retractable Mobile Roof for Large Spaces). 6: An Air Beam as in claim 4, wherein additional devices, comprising: an End-Structure at each end portion of said air beam, comprising: a Pair of Triangle Shaped Plates made of solid material (for example, nylon reinforced fiberglass), connected through a Hinge, openable as a book; One or Two Pieces of Durable Membrane installed between the two said solid plates, forming as a Cone, for the functions of connecting, sealing and freely moving along with the air beam inflation and deflation; a Steel Block (or Sliding Block), connected with the said two solid plates through the hinge, onto which (the steel block), a Steel Wheel with ball bearing, and a Sliding Ditch with cylindrical roller bearing are installed for the function of that, the end-structure will be able to smoothly slide along a Sliding Track and a Sliding Bar, and the end-structure will be able to keep certain correct orientation; an Air Valve, a Motor with wire for opening and closing the air valve, and an Air Duct installed on one of the said four solid plates for each said air beam, for the function of inflation and deflation operations; is initiated and established by inventor Keh C. Liu (Jun. 2, 2006 filed at National Patent Office of China, Beijing, File No. 200610083730.8, titled “Air Beam and Inflatable, Retractable Mobile Roof for Large Spaces).
 7. A (completely) Retractable Air Beam Roof, comprising: a Number (obtained from the length of the space, which needs roof covering, and the diameter of the air beam, which can be obtained from the results in claim 5) of the Air Beams, equally divided into two Groups; each said Group of Air Beams, paralleled and connected in certain way to form a half of the roof for large spaces, such as stadiums; Two Group of Air Beams, forming the entire roof of a large space, which can be simultaneously retracted apart towards to the opposite ends of the space (stadium), during the retraction operation; a Rain Strip (made of soft membrane) installed (firmly stuck) onto each pair of neighboring air beams for rain and snow proof. a Pair of Rain Gutters installed on the front side of top air beam of each group of air beams, coupling together when the inflation and deployment process completed; two Straight Line Shaped and Horizontal Sliding Tracks and Sliding Bars, installed to match the end-structures of the air beams as in claim 6, for smoothly sliding operation; two End Portions of each said sliding track and sliding bar, bended down within the same vertical plane, forming a 90 degree circular arc, keeping the same incline orientation (optimum angle about 30 degree from the vertical direction), for the deflated air beams be neatly stacked; four Main Winding Machines and four Pulling Chains (with length of about 120 meter each, and capacity about 1.5 ton), functioning simultaneously, needed for retraction of the air beam roof during the successively deflating the air beams, starting from the bottom one in each group of air beams; two Inflators (Inflation Machines), installed at each end, but at only one side, of the stadium, needed for the successively inflation of all the air beams, starting from the top air beam in each group simultaneously, until the said pair of rain gutters coupled, and deployment of the air beam roof completed, operating the inflators inversely also be used for speeding up the deflation and retraction process; two Main Air Ducts (with length of about 100 meter each), leading from the two inflation machines, through T-joints to the valves of every air beam, for both inflation and deflation operations; is initiated and established by inventor (Jun. 2, 2006 filed at National Patent Office of China, Beijing, File No. 200610083730.8, Titled “Air Beam and Inflatable, Retractable Mobile Roof for Large Spaces).
 8. A air beam as in claim 4, and a retractable air beam roof as in claim 7, wherein additional devices and methods (measures) needed for Strengthening the Ability of Wind Resistance of the said retractable air beam roof, comprising: i) Math derivation and Physics analysis, resulting the Optimum Arc Angle of the Air Beams about 120 degree for reaching the strongest wind resistance ability; ii) eight Nylon Reinforced Ropes installed on each air beam, four on each side, with an apart distance of about double the diameter of the air beam (tube), leading to the end-structure at the other side of the air beam, with adjustable, variable tension forces along with the wind speed; iii) a hard rib strip, made of nylon reinforced fiberglass, installed along the inner (lower) side of each air beam, with the same length as the air beam, for the function of strengthening the wind resistance ability of the air beam, and for connection of nylon ropes; iv) doubly strengthened membrane material used for the area near the end-structure of each air beam due to the effect of stress concentration occurring during the weather of strong wind; is initiated and established by inventor Keh C. Liu (Jun. 2, 2006 filed at National Patent Office of China, Beijing, File No. 200610083730.8, Titled “Air Beam and Inflatable, Retractable Mobile Roof for Large Spaces).
 9. A Method of Math and Phys Derivation for Physical Characteristics as in claim 14, wherein additional methods, comprising: a method of finding the total pulling force needed to pull each side of an entire half of the air beam roof when the deflation and retraction just start, comprising: simulating the operation of a locomotive pulling a train; and supposing that, the system of chain pulling (sliding) the end-structures has a much worse friction condition, i.e. a much larger friction coefficient, than that of a train system, say ten times bigger for a conservative estimation; calculating and resulting that, an ordinary winding machine with a regular capability will be able to handle it; calculating the mutual pressure forces between the neighboring air beams in various weather conditions, and their effects to the air pressure inside the air tubes and to the supporting forces at the end-structures; are initiated and established by inventor.
 10. A Retractable Air Beam Roof as in claim 7, wherein additional devices, comprising: a Supporting Structure made of angle steel for firmly supporting each of the sliding tracks and the sliding bars, with its form and size closely depended on the basis and location, where the air beam roof is built; two pieces of flat, dry and absolutely clean Ground or Platform at the two end-area of the stadium, for placing the retracted, deflated and stacked air beams, securely guarded; an Air Duct Guider and a Wire Guider installed on the end-structure at the air valve side of each air beam, for neatly collection of the air duct and electrical wires during the deflation and retraction process; are initiated and established by inventor.
 11. A Retractable Air Beam Roof as in claim 7, wherein additional devices and methods, comprising: four Additional Winding Machines installed at the area at two ends of the stadium (two at each end area), for retraction of the air beam roof needs, operating in case of severe weather conditions, extremely low temperature, and membrane material stepping to aging (although the said device does not need in ordinary cases, it is still set in operation in all the cases for the reason of security, reliability and convenience); four pieces of Nylon Rope around the upper portion of the top air beam, two pieces for each group of air beams, leading to the said additional winding machines, with length of about 260 meter for each piece of rope; a number of Open Soft Rope Guiders, made of soft, smooth, flexible fiberglass nylon reinforced material, firmly stuck onto the air beams, for guiding the rope, two pairs on each air beam on its top, twelve pieces on each of the top air beams on its top and upper front side; twenty pieces of Closed Soft Rope Guiders, made of soft, smooth, flexible fiberglass nylon reinforced material, firmly stuck onto the front lower side of the two top air beams, ten pieces on each top air beams, five pieces at each rope location, for the function of preventing rope dislocation; a number of Closed Hard Rope Guiders, made of hard, smooth fiberglass nylon reinforced material, for guiding the rope, two of them installed on each rib strip of each air beam, at the rope location, and two of them installed underneath the platform for the stacked deflated air beams, for the said nylon rope be smoothly moving through; membrane material of Double Durability used for the top air beam of each group of air beams; calculating and estimating the rope pulling forces needed, closely relied on the weather conditions (wind, snow and temperature) and degree of membrane material aging, from zero to 800 kg for each winding machine; are initiated and established by inventor.
 12. A Retractable Air Beam Roof as in claim 7, wherein additional devices, comprising an Automatic Air Leaking Detecting Machine by using the method of chemical smoke, which is able to find which air beam and which portion there exists a leaking hole, so that, the repair work can be quickly proceeded in time; an Automatic Snow Removing Machine, installed at the area, where the deflated air beam stacked; an Automatic Dust and Stain Cleaning and Moisture Drying Machine, installed at the area, where the deflated air beams stacked; are initiated by inventor.
 13. A Retractable Air Beam Roof as in claim 7, wherein additional devices, comprising: a Central Control Table and an Automatic Central Control Panel for automatic control of the operation of entire Retractable Air Beam Roof an Electrical Wire System, connecting all the motors (controlling valves) on each air beam with the central control table, and connecting all the eight winding machines and the two inflation machines with the central control table; wirelessly connecting the local weather forecast station with the central control table; wirelessly connecting the local police station with the central control table; wirelessly connecting the headquarter office with the central control table; an electrical wire system, connecting the loudspeaker system to spectators and players with the central control table; Programming the entire procedures starting from the weather forecast until completion of roof deployment or retraction, for the purpose of automatic operation; are initiated and established by inventor (Jun. 2, 2006 filed at National Patent Office of China, Beijing, File No. 200610083730.8, Titled “Air Beam and Inflatable, Retractable Mobile Roof for Large Spaces). 